00:01
Hello everyone we are going to solve a question and this question we are given d square y divided by dt square plus 2 dy divided by dt minus 15 y is equals to e raised to the part 3t minus t square and we have to find the solution of this equation.
00:20
So writing the auxiliary equation for the given equation we have auxiliary equation.
00:29
We have auxiliary equation as m square plus 2m minus 15 is equals to 0.
00:39
When we factorize this equation, we have m plus 5 multiplied with m minus 3 is equal to 0.
00:46
So we have m is minus 5 and 3.
00:50
So from here we can write that yc, that is the complement solution as equals to c1, e to 1, eras to power minus 5 t, plus c2 e raised to the plus c2, power 3 t so now for finding the particular solution of this we have d square plus 2d minus 15 multiplied with y is equals to e raised to the power 3 t minus t square so from here we have the particular solution is equals to e raised to the power 3 t divided by d square plus 2d minus 15 and from here we have minus t squared divided by d squared plus 2d minus 15.
01:35
On factorizing, we can write this as particular solution is equals to e raised to the part 3t divided by d plus 5 and this multiplied with d minus 3 and minus t squared divided by d plus 5 multiplied with d minus 3.
01:55
So now for finding the particular solution we have that this y particular that is y p is equals to e raised to the power 3 t on putting the value of d as 3 we get 8 d minus here we have 1 divided by 8 and from here we can write this as 1 divided by d minus 3 minus 1 divided by d minus 3 minus 1 divided by d plus 5 and this multi with t squared.
02:27
Now on solving this we have this y particular is equals to t e raised to the part 3 t divided by 8 and minus 1 divided by 8.
02:38
Here we have minus 1 divided by 3 and here 1 minus d raised to the power minus 1...